Auxiliary Prognostic-State Transport

ERF contains a generic transport substrate for prognostic atmospheric state that is owned outside the model’s core conserved-state array. The purpose of this layer is to let a separately owned constituent system use ERF’s actual mass carrier, mapped geometry, and host time integration without reproducing those parts of the dynamical core.

Note

The auxiliary-state layer is an implementation interface, not a user-selectable generic tracer package. Its qualification consumer remains a test-only inert tracer. Spectral-bin microphysics is also a live user of the same mapped-measure, host-stage, dry-air-carrier, mapped-divergence, and completed-step-transfer contracts. SBM supplies its own particle-bin semantics, coupled transport acceptance, canonical-state checks, cloud/rain projection, restart, and boundary policy.

Why auxiliary state exists

Some atmospheric constituent systems naturally contain many more prognostic components than belong in ERF’s compact core state. Examples include sectional or spectral particle populations and, potentially, other multi-component constituent systems. Keeping such state in separately owned MultiFab objects is useful only if its physical-space transport remains consistent with the atmospheric flow that transports ERF’s native scalars.

The ownership boundary is therefore intentional:

• ERF supplies the host stage timing, dry-air density, mapped geometry, and the stage carrier used by native scalar transport.

• The generic auxiliary layer supplies the mapped conservation and time-integration seam for separately owned components.

• Each scientific consumer defines the meaning and units of its components, constructs its accepted face-transfer rates, and owns its physical source terms, constraints, boundary conditions, AMR and restart behavior, and coupling projections.

The generic component metadata record a schema identifier together with a name, semantic identifier, and units for each component. The mapped transport layer deliberately does not interpret those scientific meanings.

Density-weighted state and mapped conservation

Consider an auxiliary atmospheric quantity whose intensive value per unit mass of dry air is \(z_a\). Its density-weighted state is

\[U_a = \rho_d z_a,\]

where \(\rho_d\) is the dry-air density. The generic transport routines advance \(U_a\); the intensive value is reconstructed from a state and its matching density only when an operator such as advection requires it.

For the mapped-coordinate convention used by ERF, the cell measure is

\[\omega = \frac{\det J}{m_x m_y},\]

where \(\det J\) is the cell-centered mapping Jacobian and \(m_x\) and \(m_y\) are the horizontal map factors. The corresponding mapped conservative state is

\[H_a = \omega U_a.\]

The current measure builder requires \(\det J\), \(m_x\), \(m_y\), and \(\omega\) to be finite and strictly positive in every valid cell. It also requires the metric fields to have the expected ERF cell and horizontal-map layouts. A metric or layout failure is rejected rather than replaced by a default measure.

Mapped face-transfer convention

The generic stage update consumes a face-centered \(\widetilde{\mathbf F}_a\) that is already expressed in ERF’s mapped face-transfer convention. Its computational-coordinate divergence is

\[D_\xi\left(\widetilde{\mathbf F}_a\right) = \sum_{d=1}^{3} \frac{ \widetilde F_{a,d,+} - \widetilde F_{a,d,-} }{\Delta \xi_d}.\]

No additional physical face-area factor and no second application of the map factors belongs in this divergence. The geometric factors have already entered through the mapped face-transfer construction and the cell measure \(\omega\).

The reusable divergence/application routine is agnostic to how a valid mapped face-transfer rate was produced. For example, the qualification tests feed it both native scalar-advection transfers and mapped transfers derived from ERF’s native scalar-diffusion operators. The current live inert-tracer consumer, however, constructs advection only; scalar diffusion is deliberately disabled in that fixture.

Semantic time views and the stage update

Auxiliary transport follows the host integrator and does not advance on an independent timestep. Each stage context identifies three semantic states:

• the anchor state at the old-step time;

• the input state from which the current stage transfer is constructed;

• the target state produced by the current stage.

State, dry-air density, and mapped measure are each carried with these explicit time identities. This distinction matters when density or geometry differs between stages. In particular, an intensive quantity must not be formed from an input-stage auxiliary state and a density belonging to a different stage.

For recurrence weights \(a\) and \(b\) and a face-rate time coefficient \(c\), the generic mapped update is exactly

\[H_a^{\mathrm{target}} = a\,\omega^{\mathrm{anchor}} U_a^{\mathrm{anchor}} + b\,\omega^{\mathrm{input}} U_a^{\mathrm{input}} - c\,D_\xi\left(\widetilde{\mathbf F}_a\right),\]

followed by

\[U_a^{\mathrm{target}} = \frac{H_a^{\mathrm{target}}} {\omega^{\mathrm{target}}}.\]

The mutable target storage is required to be disjoint from both read-only input states. The anchor and input views may intentionally refer to the same storage when the host recurrence permits it. Keeping the target separate prevents an in-place stage update from becoming dependent on traversal order when a future transfer construction or limiter uses neighboring values.

Carrier mass flux and intensive transport variables

When advection is written for an intensive constituent, the input-stage value is formed from matching semantic views,

\[z_a^{\mathrm{input}} = \frac{U_a^{\mathrm{input}}} {\rho_d^{\mathrm{input}}}.\]

The qualification fixture then passes this intensive field to ERF’s native scalar-advection operator together with the host stage carrier. Schematically, its mapped constituent face-transfer rate is

\[\widetilde F_{a,f} = \widetilde F_{d,f}\,z_{a,f},\]

where \(\widetilde F_{d,f}\) denotes the carrier supplied by ERF at that stage and \(z_{a,f}\) is the native reconstructed face value.

The auxiliary layer does not reconstruct a separate carrier by multiplying a sampled density and velocity. Using the host carrier is part of the coupling contract.

One important consistency check is preservation of a constant constituent ratio. If initially

\[U_a = k\rho_d\]

for a spatially constant \(k\), then \(z_a=k\). When the auxiliary state and dry-air density are advanced with the same carrier and compatible host recurrence, the discrete transport should preserve

\[U_a^{\mathrm{new}} = k\rho_d^{\mathrm{new}}\]

apart from floating-point roundoff. The auxiliary qualification tests verify this property with spatially varying density and stage-dependent carrier fields.

Face-flux rates and completed-step fluxes

A face-flux rate used inside a Runge–Kutta stage and the time-integrated face flux associated with a completed timestep are different numerical objects. ERF represents them separately so that predictor-stage intervals cannot be mistaken for completed-step transport.

See Time Advance for ERF’s host time integrators.

Compressible three-stage advance

Let \(\widetilde{\mathbf F}_0\), \(\widetilde{\mathbf F}_1\), and \(\widetilde{\mathbf F}_2\) be the mapped auxiliary face-flux rates at the three compressible stages. The mapped conservative state follows

\[H_a^* = H_a^n - \frac{\Delta t}{3} D_\xi\left(\widetilde{\mathbf F}_0\right),\]

\[H_a^{**} = H_a^n - \frac{\Delta t}{2} D_\xi\left(\widetilde{\mathbf F}_1\right),\]

and

\[H_a^{n+1} = H_a^n - \Delta t D_\xi\left(\widetilde{\mathbf F}_2\right).\]

The first two stages are predictors anchored at the old state. They are not three additive forward-Euler increments. Consequently, the completed-step integrated mapped face flux is

\[\mathbf J_a = \Delta t\,\widetilde{\mathbf F}_2.\]

The first two predictor rates therefore receive zero weight in the completed-step flux ledger.

Anelastic two-stage Runge–Kutta advance

For ERF’s two-stage anelastic Runge–Kutta method, the first stage is

\[H_a^* = H_a^n - \Delta t D_\xi\left(\widetilde{\mathbf F}_0\right),\]

and the second stage is

\[H_a^{n+1} = \frac{1}{2}H_a^n + \frac{1}{2}H_a^* - \frac{\Delta t}{2} D_\xi\left(\widetilde{\mathbf F}_1\right).\]

Equivalently,

\[H_a^{n+1} = H_a^n - \frac{\Delta t}{2} D_\xi\left( \widetilde{\mathbf F}_0 + \widetilde{\mathbf F}_1 \right),\]

so the completed-step integrated mapped face flux is

\[\mathbf J_a = \frac{\Delta t}{2} \left( \widetilde{\mathbf F}_0 + \widetilde{\mathbf F}_1 \right).\]

The completed-step ledger enforces stage order, a fixed integrator and old-step time across the step, compatible face layouts, and the exact temporal weights above. It supports multiple auxiliary components and applies the host time weighting to every component.

ERF also provides an anelastic midpoint integrator, but the current auxiliary stage adapter deliberately rejects that method because it has not yet been qualified for this transport interface.

Mapped-geometry verification

The generic mapped-divergence convention is tested independently against ERF’s native scalar operators. The current unit evidence includes:

• direct arithmetic checks of the computational mapped divergence;

• telescoping of an arbitrary periodic mapped face field;

• parity with native scalar-advection divergence using nontrivial Jacobian and horizontal map factors;

• parity when native scalar-diffusion transfers are supplied on the flat native grid and on a vertically stretched grid;

• parity with the static-terrain mapped-diffusion transfer, including the terrain metric cross terms;

• preservation of a constant constituent ratio through both supported host recurrences;

• exact completed-step temporal weights for one and multiple components;

• rejection of incompatible cell/face layouts, aliased stage targets, and invalid or out-of-order stage sequences.

The tests also contain metric negative controls. Omitting \(\det J\) from the cell measure, applying a horizontal map factor a second time, using a raw terrain vertical diffusive flux in place of ERF’s mapped transfer, or omitting the terrain cross terms must disagree with the corresponding native ERF result. These controls are intended to distinguish the actual ERF metric convention from an internally consistent but incorrect duplicate convention.

The operator-level tests are broader than the current live proof consumer. They demonstrate the generic mapped algebra for those supplied transfers; they do not by themselves establish a complete user-facing tracer capability for every tested geometry or transport process.

Live inert-tracer proof consumer

The only integrated ERF consumer of this substrate at the present revision is a one-component inert tracer used for qualification. It is intentionally restricted so that the test isolates the coupling seam rather than silently claiming a general tracer implementation.

The fixture requires:

• three spatial dimensions and exactly one AMR level;

• triply periodic boundaries;

• ConstantDz geometry with no terrain or buildings;

• moisture disabled and gravity disabled;

• no acoustic substepping;

• no molecular, numerical, LES, RANS, or PBL transport;

• ERF’s native scalar-advection path with centered-second-order horizontal and vertical dry-scalar advection;

• the two-stage RK method when the host run is anelastic; anelastic midpoint is rejected;

• the Scalar Advection/Diffusion ERF test problem.

Checkpoint/restart, coarse-to-fine initialization, and regrid/remake are explicitly rejected by this fixture. The fixture also requires a nonzero host carrier so that the live test actually exercises transport.

The proof consumer uses a time-independent mapped measure. Its storage is defined first, but \(\omega\) is built only after ERF has finalized the level Jacobian and horizontal map factors. Initialization and stage advancement fail if that measure has not been successfully built. Its old-step, input, and target measure views intentionally refer to the same static field. A moving terrain consumer would need distinct geometry at the relevant semantic times and a separate geometric-conservation qualification.

During the live smoke test, the tracer uses the host stage carrier, produces stage-varying mapped face rates, checks finite state, and closes the completed step against the integrated face-flux ledger to roundoff. These diagnostics are qualification evidence for the fixture; they are not global reductions that every future production consumer is required to perform on every stage.

Current scope and consumer responsibilities

The reusable auxiliary-state layer is deliberately narrower than a complete constituent model. In particular, the current generic substrate does not by itself provide:

• a user-facing runtime registry for arbitrary tracers;

• scientific source or sink terms;

• application-specific positivity, realizability, or composition constraints;

• physical boundary-condition policies for a new constituent family;

• AMR prolongation, restriction, reflux, or restart semantics for a new consumer;

• a qualified moving-terrain, embedded-boundary, or acoustically substepped auxiliary advance;

• a universal diffusion model or turbulence closure;

• a consumer-specific projection into ERF core coupling fields.

A new consumer must define and qualify those responsibilities explicitly. It must also ensure that any supplied face-transfer rate follows ERF’s mapped convention and that its state, density, measure, and stage timing are paired consistently.

The generic architecture is intended to make such consumers possible without reimplementing ERF’s carrier, geometry, or time-integration mathematics. A sectional aerosol system, atmospheric chemistry state, specialized tracer family, or another multi-component distribution could in principle use this interface, but none of those possibilities is advertised here as an existing runtime capability.

Relationship to spectral-bin microphysics

The SBM spectral state is owned separately from ERF’s core conserved-state array. SBMTransport uses the generic mapped auxiliary-state interfaces for the static mapped measure, host-stage timing, dry-air mass carrier, conservative transport update, and accepted completed-step accounting. SBM retains ownership of spectral-bin coordinates, mass/number admissibility, and cloud/rain projection. The inert tracer remains a small proof consumer of the shared transport infrastructure.

The code also contains independently tested SBM-specific operators for conservative mapped restriction, dry-air-relative prolongation, explicit old/new spectral time views, and an all-component accepted-transfer record. These methods are not yet connected to the complete adaptive-grid lifecycle. Operational SBM simulations remain single-level and triply periodic; coarse/fine time interpolation, flux correction, regridding, multilevel restart, and nonperiodic spectral boundaries are not yet available.

The auxiliary layer addresses transport between atmospheric grid cells. It does not define how an SBM particle distribution is represented or remapped in particle-mass space, nor does it provide SBM realizability limiting, sedimentation, activation, collision-coalescence, or other cloud microphysics. Those scientific contracts are documented separately in Spectral-Bin Microphysics (SBM).